Existence, Uniqueness, and Asymptotic Behavior for Nonlocal Parabolic Problems with Dominating Gradient Terms

In this paper we deal with the well-posedness of Dirichlet problems associated to nonlocal Hamilton--Jacobi parabolic equations in a bounded, smooth domain $\Omega$, in the case when the classical boundary condition may be lost. We address the problem for both coercive and noncoercive Hamiltonians: for coercive Hamiltonians, our results rely more on the regularity properties of the solutions, while noncoercive case are related to optimal control problems and the arguments are based on a careful study of the dynamics near the boundary of the domain. Comparison principles for bounded sub- and supersolutions are obtained in the context of viscosity solutions with generalized boundary conditions, and consequently we obtain the existence and uniqueness of solutions in $C(\bar{\Omega} \times [0,+\infty))$ by the application of Perron's method. Finally, we prove that the solution of these problems converges to the solutions of the associated stationary problem as $t \to +\infty$ under suitable assumptions on the...

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